I’ve just learned Schweizer, which strikes me as unusual for an adjective in:
- having a capital letter;
- not being inflected for case/gender/number.
Do the two always go hand-in-hand, and is Schweizer a rare case?
Adjectives derived from geographical terms ending in -er are always capitalized and not inflected. There are quite a few of these adjectives (especially for towns: Berliner, Hamburger, Münchner, Bremer, Kölner…), but I'm not aware of a general rule that tells us whether geographical adjectives are formed using -er (vs. -isch etc.).
For clarification, if it is not exactly clear what the OP means:
There are adjectives which don’t have capital letters and which don’t inflect – such as rosa – so no, the two don’t always go hand-in-hand.
If it is part of a fixed name like "Kölner Dom", "der Regierende Bürgermeister" the adjective is capitalized... here is the source
Those DO inflect though...
An den Regierenden Bürgermeister
Der Regierende Bürgermeister von Berlin ist Klaus W.
Klaus W., Regierender Bürgermeister von Berlin, ist in der SPD.